Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. \(\log _{3}\left(\frac{1}{27}\right)\)

Step-by-Step Solution

Answer

-3

Step 1: Understand the Logarithm

A logarithm \( \log_b a\ \) is the exponent to which the base \( b \) must be raised to get \( a \). Here, we are finding what exponent \( x \) makes \( 3^x = \frac{1}{27} \).

Step 2: Express Base in Known Powers

Recognize that \( 27 = 3^3 \), so \( \frac{1}{27} = \frac{1}{3^3} = 3^{-3} \). Now the problem is simplified to finding the exponent \( x \) so that \( 3^x = 3^{-3} \).

Step 3: Solve for the Exponent

Since the bases are the same, set the exponents equal: \( x = -3 \). This is done under the property that if \( b^m = b^n \), then \( m = n \). Hence, \( x = -3 \).

Key Concepts

Exponents

Exponents are a cornerstone of mathematics, frequently used both in basic and advanced math exercises. An exponent refers to the number of times a base is multiplied by itself. For example, in the term \(3^4\), 3 is the base and 4 is the exponent. This means 3 is multiplied by itself 4 times, and the result is 81. Exponents make multiplication more straightforward and allow us to express large numbers succinctly.Understanding exponents is crucial for working with logarithms, as they essentially ask, "To what exponent must we raise this base to get a specific number?" In logarithms, knowing exponents allows us to translate between exponential form and logarithmic form seamlessly.

Base Conversion

Base conversion involves changing numbers from one number base to another. Although primarily used in computer science, understanding base conversion is essential for evaluating logarithms. In logarithms, the base is crucial because it determines how we express exponents.For example, \(\log_3 (27)\) is all about determining what power you raise 3 to get 27. If we solve it, we find that \(3^3 = 27\), so this logarithm evaluates to 3. However, in base conversion between different types of bases, like binary, decimal, or hexadecimal, the base numbers will be different, yet the concept of calculating power stays fundamentally the same.

Negative Exponents

Negative exponents might seem tricky at first, but they follow a straightforward rule: a negative exponent indicates that you take the reciprocal of the base raised to the corresponding positive exponent. For instance, \(3^{-3}\) is the reciprocal of \(3^3\). So what you do is flip \(3^3\) upside down.Breaking it down, \(3^3 = 27\), which means \(3^{-3} = \frac{1}{27}\). Thus, negative exponents simply represent how exponents indicate division or fractions instead of just multiplication. When you see a logarithm like \(\log_3 (\frac{1}{27})\), recognizing that \(\frac{1}{27} = 3^{-3}\) helps to solve it easily.

Properties of Logarithms

Logarithms have several key properties that are helpful in simplifying complex expressions and performing calculations without a calculator.

• Product Property: \(\log_b (mn) = \log_b m + \log_b n\). This property translates multiplication within the logarithmic function into addition outside of it.
• Quotient Property: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\). It turns division into subtraction, aiding simplification.
• Power Property: \(\log_b (m^n) = n \cdot \log_b m\). Exponents inside the logarithm become coefficients, making heavily nested problems simpler.

Applying these properties enables us to tackle a variety of logarithmic problems, including those involving base conversions or negative exponents, with ease.